From e9cbdfede2cde13865ce97b23f6635de99ed21de Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 31 May 2021 07:18:23 +0200 Subject: [PATCH] saving work for lecture 2 --- 2021/Lecture_2/ISTPC_Loos_2.tex | 942 ++++++++++++++++++++++++-------- 2021/Lecture_2/fig/BSE-GW.tex | 12 +- 2 files changed, 706 insertions(+), 248 deletions(-) diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex index a8903a4..66bd371 100644 --- a/2021/Lecture_2/ISTPC_Loos_2.tex +++ b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -26,13 +26,19 @@ } \definecolor{darkgreen}{RGB}{0, 180, 0} -\newcommand{\red}[1]{\textcolor{red}{#1}} -\newcommand{\purple}[1]{\textcolor{purple}{#1}} +\definecolor{fooblue}{RGB}{0,153,255} +\definecolor{fooyellow}{RGB}{234,180,0} +\definecolor{lavender}{rgb}{0.71, 0.49, 0.86} +\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36} +\newcommand{\violet}[1]{\textcolor{lavender}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} -\newcommand{\green}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\purple}[1]{\textcolor{purple}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} -\newcommand{\pub}[1]{\textcolor{purple}{#1}} -\newcommand{\violet}[1]{\textcolor{violet}{#1}} +\newcommand{\green}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}} +\newcommand{\red}[1]{\textcolor{red}{#1}} +\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}} +\newcommand{\pub}[1]{\small \textcolor{purple}{#1}} \newcommand{\cdash}{\multicolumn{1}{c}{---}} \newcommand{\mc}{\multicolumn} @@ -40,20 +46,30 @@ \newcommand{\mr}{\multirow} \newcommand{\br}{\bm{r}} \newcommand{\ree}{r_{12}} +\newcommand{\T}[1]{#1^{\intercal}} % methods -\newcommand{\evGW}{evGW} -\newcommand{\qsGW}{qsGW} -\newcommand{\scGW}{scGW} -\newcommand{\GOWO}{G$_0$W$_0$} -\newcommand{\GOW}{G$_0$W} -\newcommand{\GWO}{GW$_0$} -\newcommand{\GW}{GW} +\newcommand{\evGW}{ev$GW$} +\newcommand{\qsGW}{qs$GW$} +\newcommand{\scGW}{sc$GW$} +\newcommand{\GOWO}{$G_0W_0$} +\newcommand{\GOW}{$G_0W$} +\newcommand{\GWO}{$GW_0$} +\newcommand{\GW}{$GW$} \newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX} \newcommand{\GWSOSEX}{{\GW}+SOSEX} -\newcommand{\GnWn}[1]{G$_{#1}$W$_{#1}$} -\newcommand{\GOF}{G$_0$F2} -\newcommand{\GF}{GF2} +\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$} +\newcommand{\GOF}{$G_0F2$} +\newcommand{\GF}{$GF2$} +\newcommand{\KS}{\text{KS}} +\newcommand{\RPA}{\text{RPA}} +\newcommand{\RPAx}{\text{RPAx}} +\newcommand{\BSE}{\text{BSE}} +\newcommand{\TDA}{\text{TDA}} +\newcommand{\xc}{\text{xc}} +\newcommand{\Ha}{\text{H}} +\newcommand{\co}{\text{c}} +\newcommand{\x}{\text{x}} % operators \newcommand{\hH}{\Hat{H}} @@ -88,7 +104,8 @@ \newcommand{\eGOF}[1]{\epsilon^\text{\GOF}_{#1}} \newcommand{\de}[1]{\Delta\epsilon_{#1}} \newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}} -\newcommand{\Om}[1]{\Omega_{#1}} +\newcommand{\deKS}[1]{\Delta\epsilon^\text{KS}_{#1}} +\newcommand{\Om}[2]{\Omega_{#1}^{#2}} \newcommand{\eHOMO}{\epsilon_\text{HOMO}} \newcommand{\eLUMO}{\epsilon_\text{LUMO}} @@ -97,8 +114,10 @@ % Matrix elements -\newcommand{\A}[1]{A_{#1}} -\newcommand{\B}[1]{B_{#1}} +\newcommand{\A}[2]{A_{#1}^{#2}} +\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}} +\newcommand{\B}[2]{B_{#1}^{#2}} +\newcommand{\tB}[2]{\Tilde{B}_{#1}^{#2}} \renewcommand{\S}[1]{S_{#1}} \newcommand{\ABSE}[1]{A^\text{BSE}_{#1}} \newcommand{\BBSE}[1]{B^\text{BSE}_{#1}} @@ -113,6 +132,7 @@ \newcommand{\vc}[1]{v_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} +\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} \newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} \newcommand{\SigGWSOSEX}[1]{\Sigma^\text{\GWSOSEX}_{#1}} \newcommand{\SigGF}[1]{\Sigma^\text{\GF}_{#1}} @@ -141,15 +161,13 @@ \newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}} \newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}} -% Wigner symbols -\newcommand{\WJ}[3]{ -\begin{pmatrix} -#1 & #2 & #3 \\ -0 & 0 & 0 \\ -\end{pmatrix} -} -\newcommand{\ERI}[3]{\qty(#1 #2 #3)} -\newcommand{\sERI}[3]{\qty[#1 #2 #3]} +\newcommand{\MO}[1]{\phi_{#1}} +\newcommand{\ERI}[2]{(#1|#2)} +\newcommand{\rbra}[1]{(#1|} +\newcommand{\rket}[1]{|#1)} +\newcommand{\sERI}[2]{[#1|#2]} +\newcommand{\sig}{\sigma} +\newcommand{\sigp}{\sigma'} % Matrices \newcommand{\bF}{\bm{F}} @@ -165,7 +183,9 @@ \newcommand{\be}{\bm{\epsilon}} \newcommand{\bDelta}{\bm{\Delta}} \newcommand{\beHF}{\bm{\epsilon}^\text{HF}} +\newcommand{\beKS}{\bm{\epsilon}^\text{KS}} \newcommand{\bcHF}{\bm{c}^\text{HF}} +\newcommand{\bcKS}{\bm{c}^\text{KS}} \newcommand{\beGW}{\bm{\epsilon}^\text{\GW}} \newcommand{\beGnWn}[1]{\bm{\epsilon}^\text{\GnWn{#1}}} \newcommand{\bcGnWn}[1]{\bm{c}^\text{\GnWn{#1}}} @@ -174,12 +194,20 @@ \newcommand{\bdeHF}{\bm{\Delta\epsilon}^\text{HF}} \newcommand{\bdeGW}{\bm{\Delta\epsilon}^\text{GW}} \newcommand{\bdeGF}{\bm{\Delta\epsilon}^\text{GF2}} -\newcommand{\bOm}{\bm{\Omega}} -\newcommand{\bA}{\bm{A}} -\newcommand{\bB}{\bm{B}} -\newcommand{\bX}{\bm{X}} -\newcommand{\bY}{\bm{Y}} -\newcommand{\bZ}{\bm{Z}} +\newcommand{\bO}{\bm{0}} +\newcommand{\bI}{\bm{1}} +\newcommand{\bOm}[2]{\bm{\Omega}_{#1}^{#2}} +\newcommand{\bA}[2]{\bm{A}_{#1}^{#2}} +\newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}} +\newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}} +\newcommand{\bB}[2]{\bm{B}_{#1}^{#2}} +\newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} +\newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}} +\newcommand{\bK}[2]{\bm{K}_{#1}^{#2}} +\newcommand{\bP}[2]{\bm{P}_{#1}^{#2}} + +\newcommand{\yo}{\yellow{\omega}} \newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}} @@ -221,24 +249,40 @@ decoration={snake, \begin{document} -%%% SLIDE 1 %%% +%----------------------------------------------------- \begin{frame} \titlepage \end{frame} -% +%----------------------------------------------------- -%%% SLIDE 2 %%% +%----------------------------------------------------- \begin{frame}{Today's program} \begin{itemize} - \item + \item Charged excitations: + \begin{itemize} + \item One-shot $GW$ (\GOWO) + \item Partially self-consistent $GW$ (\evGW) + \item Self-consistent $GW$ (\qsGW) + \item $GW$ vs GF + \end{itemize} + \item Neutral excitations + \begin{itemize} + \item Configuration interaction with singles (CIS) + \item Time-dependent Hartree-Fock (TDHF) + \item Random-phase approximation (RPA) + \item Time-dependent density-functional theory (TDDFT) + \item Bethe-Salpeter equation (BSE) formalism + \end{itemize} + \item Total energies + \begin{itemize} + \item Plasmon formula + \item Galitski-Migdal formulation + \item Adiabatic connection fluctuation-dissipation theorem (ACFDT) + \end{itemize} \end{itemize} \end{frame} -% +%----------------------------------------------------- -%----------------------------------------------------- -\section{Theory} -%----------------------------------------------------- -\subsection{Hedin's pentagon} %----------------------------------------------------- \begin{frame}{Hedin's pentagon} \begin{columns} @@ -249,47 +293,635 @@ decoration={snake, \pub{Hedin, Phys Rev 139 (1965) A796} \end{column} \begin{column}{0.6\textwidth} - \begin{block}{What can you calculate with GW?} + \begin{block}{What can you calculate with $GW$?} + \begin{itemize} + \item Ionization potentials (IPs) given by occupied MO energies + \item Electron affinities (EAs) given by virtual MO energies + \item Fundamental (HOMO-LUMO) gap (or band gap in solids) + \item Correlation and total energies + \end{itemize} + \end{block} + \begin{block}{What can you calculate with BSE?} \begin{itemize} - \item Ionization potentials (IP) given by occupied MO energies - \bigskip - \item Electron affinities (EA) given by virtual MO energies - \bigskip - \item HOMO-LUMO gap (or band gap in solids) - \bigskip \item Singlet and triplet neutral excitations (vertical absorption energies) - \bigskip + \item Oscillator strengths (absorption intensities) \item Correlation and total energies \end{itemize} \end{block} \end{column} \end{columns} \end{frame} +%----------------------------------------------------- %----------------------------------------------------- -\subsection{GW flavours} +\begin{frame}{Fundamental and optical gaps} + \begin{center} + \includegraphics[width=\textwidth]{fig/gaps} + \end{center} +\end{frame} %----------------------------------------------------- -\begin{frame}{GW flavours} + +%----------------------------------------------------- +\begin{frame}{The MBPT chain of actions} + \begin{center} + \includegraphics[width=0.7\textwidth]{fig/BSE-GW} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{$GW$ flavours} \begin{block}{Acronyms} \begin{itemize} \bigskip - \item perturbative GW one-shot GW, or \green{\GOWO} + \item perturbative $GW$, one-shot $GW$, or \green{\GOWO} \bigskip - \item \orange{\evGW} or eigenvalue-only (partially) self-consistent GW + \item \orange{\evGW} or eigenvalue-only (partially) self-consistent $GW$ \bigskip - \item \red{\qsGW} or quasiparticle (partially) self-consistent GW + \item \red{\qsGW} or quasiparticle (partially) self-consistent $GW$ \bigskip - \item \violet{\scGW} or (fully) self-consistent GW - \bigskip - \item \purple{BSE} or Bethe-Salpeter equation for neutral excitations + \item \violet{\scGW} or (fully) self-consistent $GW$ \bigskip \end{itemize} \end{block} \end{frame} %----------------------------------------------------- -\subsection{Literature} + %----------------------------------------------------- -\begin{frame}{useful papers for chemists} +\begin{frame}{Green's function and dynamical screening} + \begin{block}{One-body Green's function} + \begin{equation} + \blue{G}(\br_1,\br_2;\yo) + = \sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta} + + \sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta} + \end{equation} + \end{block} + \begin{block}{Non-interacting polarizability} + \begin{equation} + P(\br_1,\br_2;\omega) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\omega+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega' + \end{equation} + \end{block} + \begin{block}{Dielectric function} + \begin{equation} + \epsilon(\br_1,\br_2;\omega) = \delta(\br_1 - \br_2) - \int \frac{P(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 + \end{equation} + \end{block} + \begin{block}{Dynamically-screened Coulomb potential} + \begin{equation} + \highlight{W}(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Dynamical screening in a basis} + \begin{block}{Spectral representation of $W$} + \begin{equation} + \begin{split} + \highlight{W}_{pq,rs}(\yo) + & = \iint \MO{p}(\br_1) \MO{q}(\br_1) \highlight{W}_{pq,rs}(\br_1,\br_2;\yo) \MO{r}(\br_2) \MO{s}(\br_2) d\br_1 d\br_2 + \\ + & = \underbrace{\ERI{pq}{rs}}_{\text{(static) exchange part}} + + \underbrace{2 \sum_m \violet{\ERI{pq}{m}} \violet{\ERI{rs}{m}} + \qty[ \frac{1}{\yo - \orange{\Om{m}{\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\RPA}} - i \eta} ]}_{\text{(dynamical) correlation part } \highlight{W}^{\co}_{pq,rs}(\yo)} + \end{split} + \end{equation} + \end{block} + \begin{block}{Electron repulsion integrals (ERIs)} + \begin{equation} + \ERI{pq}{rs} = \iint \frac{\MO{p}(\br_1) \MO{q}(\br_1) \MO{r}(\br_2) \MO{s}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2 + \end{equation} + \end{block} + \begin{block}{Screened ERIs (or spectral weights)} + \begin{equation} + \violet{\ERI{pq}{m}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\RPA}+\bY{m}{\RPA}})_{ia} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Computation of the dynamical screening} + \begin{block}{Direct RPA calculation (pseudo-hermitian linear problem)} + \begin{equation} + \begin{pmatrix} + \bA{}{} & \bB{}{} \\ + -\bB{}{} & -\bA{}{} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \orange{\bX{m}{}} \\ + \orange{\bY{m}{}} \\ + \end{pmatrix} + = + \orange{\Om{m}{}} + \begin{pmatrix} + \orange{\bX{m}{}} \\ + \orange{\bY{m}{}} \\ + \end{pmatrix} + \end{equation} + \begin{equation} + \qq*{For singlet states:} \A{ia,jb}{\RPA} = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2\ERI{ia}{bj} + \qquad + \B{ia,jb}{\RPA} = 2\ERI{ia}{jb} + \end{equation} + \end{block} + \begin{block}{Non-hermitian to hermitian} + \begin{equation} + (\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{m}{} = \Om{m}{2} \bZ{m}{} + \end{equation} + \begin{gather} + (\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{} + \\ + (\bX{}{} - \bY{}{})_m = \Om{m}{+1/2} (\bA{}{} - \bB{}{})^{-1/2} \cdot \bZ{m}{} + \end{gather} + \end{block} + \begin{block}{Tamm-Dancoff approximation (TDA)} + \begin{equation} + \bB{}{} = \bO \quad \Rightarrow \quad \bA{}{} \cdot \bX{m}{} = \Om{m}{\TDA} \bX{m}{} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{The self-energy} + \begin{block}{$GW$ Self-energy} + \begin{equation} + \underbrace{\red{\Sig{}{\xc}}(\br_1,\br_2;\yo)}_{\text{$GW$ self-energy}} + = \underbrace{\purple{\Sig{}{\x}}(\br_1,\br_2)}_{\text{\purple{exchange}}} + + \underbrace{\red{\Sig{}{\co}}(\br_1,\br_2;\yo)}_{\text{\red{correlation}}} + = \frac{i}{2\pi} \int \blue{G}(\br_1,\br_2;\yo+\omega') \highlight{W}(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega' + \end{equation} + \end{block} + \begin{block}{Exchange part of the (static) self-energy} + \begin{equation} + \purple{\Sig{pq}{\x}} = - \sum_{i} \ERI{pi}{iq} + \end{equation} + \end{block} + \begin{block}{Correlation part of the (dynamical) self-energy} + \begin{equation} + \red{\Sig{pq}{\co}}(\yo) + = 2 \sum_{im} \frac{\violet{\ERI{pi}{m}} \violet{\ERI{qi}{m}}}{\yo - \e{i} + \orange{\Om{m}{\RPA}} - i \eta} + + 2 \sum_{am} \frac{\violet{\ERI{pa}{m}} \violet{\ERI{qa}{m}}}{\yo - \e{a} - \orange{\Om{m}{\RPA}} + i \eta} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Quasiparticle equation} + \begin{block}{Dyson equation} + \begin{equation} + \qty[ \blue{G}(\br_1,\br_2;\yo) ]^{-1} + = \underbrace{\qty[ G_{\KS}(\br_1,\br_2;\yo) ]^{-1}}_{\text{KS Green's function}} + + \red{\Sig{}{\xc}}(\br_1,\br_2;\yo) - \underbrace{\upsilon^{\xc}(\br_1)}_{\text{KS potential}} \delta(\br_1 - \br_2) + \end{equation} + \end{block} + \begin{block}{Non-linear quasiparticle (QP) equation} + \begin{equation} + \yo = \eKS{p} + \red{\Sig{pp}{\xc}}(\yo) - V_{p}^{\xc} + \qq{with} + V_{p}^{\xc} = \int \MO{p}(\br) \upsilon^{\xc}(\br) \MO{p}(\br) d\br + \end{equation} + \end{block} + \begin{block}{Linearized QP equation} + \begin{equation} + \blue{\eGW{p}} = \e{p}^{\KS} + \green{Z_{p}} [\red{\Sig{pp}{\xc}}(\e{p}^{\KS}) - V_{p}^{\xc} ] + \end{equation} + \end{block} + \begin{block}{Renormalization factor or spectral weight} + \begin{equation} + \green{Z_{p}} = \qty[ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \e{p}^{\KS}} ]^{-1} + \qq{with} 0 \le \green{Z_{p}} \le 1 + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Perturbative {\GW} with linearized solution} + \begin{block}{Linearized {\GOWO}~subroutine} + \begin{algorithmic} + \Procedure{{\GOWO}lin}{} + \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ + \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ + \State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ + \Comment{\alert{This is a $\order*{N^6}$ step!}} + \State Form screened ERIs $\violet{\ERI{pq}{m}}$ + \For{$p=1,\ldots,N$} + \State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ at $\yo = \eKS{p}$ + \State Compute renornalization factors \green{$\Z{p}$} + \State Evaluate $\blue{\eGOWO{p}} = \eKS{p} + \green{\Z{p}} \qty{ \Re[\red{\SigC{pp}}(\eKS{p})] - V_{p}^{\xc} }$ + \EndFor + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Perturbative {\GW} with graphical solution} + \begin{block}{Graphical {\GOWO}~subroutine} + \begin{algorithmic} + \Procedure{{\GOWO}graph}{} + \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ + \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ + \State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ + \Comment{\alert{This is a $\order*{N^6}$ step!}} + \State Form screened ERIs $\violet{\ERI{pq}{m}}$ + \For{$p=1,\ldots,N$} + \State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ + \State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGOWO{p}}$ via Newton's method + \EndFor + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Partially self-consistent eigenvalue $GW$} + \begin{block}{{\evGW} subroutine} + \begin{algorithmic} + \Procedure{partially self-consistent {\evGW}}{} + \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ + \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ + \State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ + \State Form screened ERIs $\violet{\ERI{pq}{m}}$ + \State Set $\blue{\beGnWn{-1}} = \beKS$ and $n = 0$ + \While{$\max{\abs{\bDelta}} < \tau$} + \For{$p=1,\ldots,N$} + \State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ + \State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGnWn{p}{n}}$ + \EndFor + \State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$ + \State $n \leftarrow n + 1$ + \EndWhile + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Quasiparticle self-consistent {\GW} (\qsGW)} + \begin{block}{{\qsGW} subroutine} + \begin{algorithmic} + \Procedure{partially self-consistent {\qsGW}}{} + \State Perform HF calculation to get $\beHF$ and $\bcHF$ \green{(optional)} + \State Set $\blue{\beGnWn{-1}} = \beHF$, $\blue{\bcGnWn{-1}} = \bcHF$ and $n = 0$ + \While{$\max{\abs{\bDelta}} < \tau$} + \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$ + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ + \State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ + \State Form screened ERIs $\violet{\ERI{pq}{m}}$ + \State Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form + $\red{\Tilde{\Sigma}^{\co}} \leftarrow \qty[ \red{\bSigC}(\blue{\beGnWn{n-1}})^\dag + \red{\bSigC}(\blue{\beGnWn{n-1}}) ]/2$ + \State Form $\purple{\Tilde{\bF}} = \bFHF + \red{\Tilde{\Sigma}^{\co}}$ + \State Diagonalize $\purple{\Tilde{\bF}}$ to get $\blue{\beGnWn{n}}$ and $\blue{\bcGnWn{n}}$ + \State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$ + \State $n \leftarrow n + 1$ + \EndWhile + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +\begin{frame}{TD-DFT and BSE in practice: Casida-like equations} + \begin{block}{Linear response problem} + \begin{equation*} + \boxed{\begin{pmatrix} + \red{\bA{}{}} & \orange{\bB{}{}} + \\ + \orange{-\bA{}{}} & \red{-\bB{}{}} + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{} + \\ + \bY{m}{} + \end{pmatrix} + = + \highlight{\Om{m}{}} + \begin{pmatrix} + \bX{m}{} + \\ + \bY{m}{} + \end{pmatrix}} + \end{equation*} + \end{block} + % + \begin{block}{Blue pill: TD-DFT within the \alert{adiabatic} approximation} + \begin{gather} + \red{A}_{ia,jb} = \qty( \varepsilon_a^\text{\violet{KS}} - \varepsilon_i^\text{\violet{KS}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} + \yellow{f}^{\yellow{xc}}_{ia,bj} + \qquad + \orange{B}_{ia,jb} = 2 \blue{(ia|jb)} + \yellow{f}^{\yellow{xc}}_{ia,jb} + \\ + \yellow{f}^{\yellow{xc}}_{ia,bj} = \iint \phi_{i}(\br{})\phi_{a}(\br{}) \frac{\delta^2 E^{xc} }{\delta\rho(\br{}) \delta\rho(\br{}')} \phi_{b}(\br{})\phi_{j}(\br{}) d\br{} d\br{}' + \end{gather} + \end{block} + % + \begin{block}{Red pill: BSE within the \alert{static} approximation} + \begin{gather} + \red{A}_{ia,jb} = \qty( \varepsilon_a^{\green{GW}} - \varepsilon_i^{\green{GW}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \purple{W}^\text{stat}_{ij,ba} + \qquad + \orange{B}_{ia,jb} = 2 \blue{(ia|jb)} - \purple{W}^\text{stat}_{ib,ja} + \\ + \purple{W}^\text{stat}_{ij,ab} \equiv \purple{W}_{ij,ab} (\omega = 0) = (ij|ab) - W^{c}_{ij,ab}(\omega = 0) + \end{gather} + \end{block} + % +\end{frame} + +\begin{frame}{TDHF and CIS: removing the correlation part} + \begin{block}{Linear response problem} + \begin{equation*} + \boxed{\begin{pmatrix} + \red{\bA{}{}} & \orange{\bB{}{}} + \\ + \orange{-\bA{}{}} & \red{-\bB{}{}} + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{} + \\ + \bY{m}{} + \end{pmatrix} + = + \highlight{\Om{m}{}} + \begin{pmatrix} + \bX{m}{} + \\ + \bY{m}{} + \end{pmatrix}} + \end{equation*} + \end{block} + % + \begin{block}{TDHF = RPA with exchange (RPAx)} + \begin{align} + \red{A}_{ia,jb} & = \qty( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\green{HF}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \yellow{(ij|ba)} + & + \orange{B}_{ia,jb} & = 2 \blue{(ia|jb)} - \yellow{(ib|ja)} + \end{align} + \end{block} + % + \begin{block}{Linear response problem within the Tamm-Dancoff approximation} + \begin{equation} + \boxed{\red{\bA{}{}} \cdot \bX{m}{} = \highlight{\Om{m}{}} \, \bX{m}{} } + \end{equation} + \end{block} + % + \begin{block}{TDHF within TDA = CIS} + \begin{equation} + \red{A}_{ia,jb} + = \qty( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\green{HF}} ) \delta_{ij} \delta_{ab} + + 2 \blue{(ia|bj)} - \yellow{(ij|ba)} + \end{equation} + \end{block} + % +\end{frame} + +%----------------------------------------------------- +\begin{frame}{Linear response} + \begin{block}{General linear response problem} + \begin{algorithmic} + \Procedure{Linear response}{} + \State Compute $\bA{}{}$ matrix at a given level of theory + \If{$\TDA$} + \State Diagonalize $\bA{}{}$ to get $\Om{m}{\TDA}$ and $\bX{m}{\TDA}$ + \Else + \State Compute $\bB{}{}$ matrix at a given level of theory + \State Diagonalize $\bA{}{} - \bB{}{}$ to form $(\bA{}{} - \bB{}{})^{1/2}$ + \State Form and diagonalize $(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2}$ + to get $\Om{m}{2}$ and $\bZ{m}{}$ + \State Compute $(\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}$ + \EndIf + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + + + +%----------------------------------------------------- +\begin{frame}{Correlation energy} + \begin{block}{RPA correlation energy: plasmon formula} + \begin{equation*} + \label{eq:Ec-RPA} + \EcRPA + = \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] + = \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} ) + \end{equation*} + \end{block} + \begin{block}{Galitskii-Migdal functional} + \begin{equation*} + \label{eq:GM} + \EcGM + = \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \red{\SigC{pq}}(\omega) \blue{\G{pq}}(\omega) e^{i\omega\eta} + = 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\e{a} - \e{i} + \orange{\Om{m}{\RPA}}} + \end{equation*} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Adiabatic connection fluctuation dissipation theorem (ACFDT)} + \begin{block}{Adiabatic connection} + \begin{equation} + \boxed{ + \Ec^\text{ACFDT} + = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda + \stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} w_k \Tr( \bK{}{} \bP{}{\lambda_k}) + } + \end{equation} + \end{block} + \begin{block}{Interaction kernel} + \begin{equation} + \bK{}{} = + \begin{pmatrix} + \btA{}{} & \btB{}{} + \\ + \btB{}{} & \btA{}{} + \end{pmatrix} + \qquad + \tA{ia,jb}{} = 2\lambda\ERI{ia}{bj} + \qquad + \tB{ia,jb}{} = 2\lambda\ERI{ia}{jb} + \end{equation} + \end{block} + \begin{block}{Correlation part of the two-particle density matrix} + \begin{equation} + \bP{}{\lambda} = + \begin{pmatrix} + \bY{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bY{}{\lambda} \cdot \T{(\bX{}{\lambda})} + \\ + \bX{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bX{}{\lambda} \cdot \T{(\bX{}{\lambda})} + \end{pmatrix} + - + \begin{pmatrix} + \bO & \bO + \\ + \bO & \bI + \end{pmatrix} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Gaussian quadrature} + \begin{block}{Numerical integration by quadrature} + \begin{equation} + \boxed{\int_a^b f(x) w(x) dx \approx \sum_k \underbrace{w_k}_{\text{weights}} f(\underbrace{x_k}_{\text{roots}})} + \end{equation} + \end{block} + \begin{block}{Quadrature rules} + \begin{center} + \begin{tabular}{llll} + \hline + Interval $[a,b]$ & Weight function $w(x)$ & Orthogonal polynomials & Name \\ + \hline + $[-1,1]$ & $1$ & Legendre $P_n(x)$ & Gauss-Legendre \\ + $(-1,1)$ & $(1-x)^\alpha(1+x)^\beta, \quad \alpha,\beta > -1$ & Jacobi $P_n^{\alpha,\beta}(x)$ & Gauss-Jacobi \\ + $(-1,1)$ & $1/\sqrt{1-x^2}$ & Chebyshev (1st kind) $T_n(x)$ & Gauss-Chebyshev \\ + $[-1,1]$ & $\sqrt{1-x^2}$ & Chebyshev (2nd kind) $U_n(x)$ & Gauss-Chebyshev \\ + $[0,\infty)$ & $\exp(-x)$ & Laguerre $L_n(x)$ & Gauss-Laguerre \\ + $[0,\infty)$ & $x^\alpha \exp(-x), \quad \alpha > -1$ & Generalized Laguerre $L_n^\alpha(x)$ & Gauss-Laguerre \\ + $(-\infty,\infty)$ & $\exp(-x^2)$ & Hermite $H_n(x)$ & Gauss-Hermite \\ + \hline + \end{tabular} + \\ + \bigskip + \url{https://en.wikipedia.org/wiki/Gaussian_quadrature} + \end{center} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{ACFDT at the RPA/RPAx level} + \begin{block}{RPA matrix elements} + \begin{equation} + \A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj} + \qquad + \B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb} + \end{equation} + \begin{equation} + \boxed{ + \Ec^\RPA + = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda + = \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] + } + \end{equation} + \end{block} + + \begin{block}{RPAx matrix elements} + \begin{equation} + \A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]}_{\tA{ia,jb}{\lambda,\RPAx}} + \qquad + \B{ia,jb}{\lambda,\RPAx} = \lambda \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ] + \end{equation} + \end{block} + +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{ACFDT at the BSE level} + \begin{block}{BSE matrix elements} + \begin{equation} + \A{ia,jb}{\lambda,\BSE} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ]}_{\tA{ia,jb}{\lambda,\BSE}} + \qquad + \B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)] + \end{equation} + \end{block} + \begin{block}{$\lambda$-dependent screening} + \begin{equation} + \highlight{W}_{pq,rs}^{\lambda}(\yo) + = \ERI{pq}{rs} + + 2 \sum_m \violet{\ERI{pq}{m}^{\lambda}} \violet{\ERI{rs}{m}^{\lambda}} + \qty[ \frac{1}{\yo - \orange{\Om{m}{\lambda,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\lambda,\RPA}} - i \eta} ] + \end{equation} + \begin{equation} + \violet{\ERI{pq}{m}^{\lambda}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\lambda,\RPA}+\bY{m}{\lambda,\RPA}})_{ia} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + + + +%----------------------------------------------------- +\begin{frame}{The bridge between TD-DFT and BSE} + \begin{center} + \begin{tabular}{lcr} + \hline + \bf \red{TD-DFT} & \bf \purple{Connection} & \bf \violet{BSE} + \\ + \hline + \\ + \red{One-point density} & & \violet{Two-point Green's function} + \\ + $\rho(1)$ & $\rho(1) = -iG(11^{+})$ & $G(12)$ + \\ + \\ + \red{Two-point susceptibility} & & \violet{Four-point susceptibility} + \\ + $\chi(12) = \pdv{\rho(1)}{U(2)}$ & $\chi(12) = -i L(12;1^+2^+)$ & $L(12;34) = \pdv{G(13)}{U(42)}$ + \\ + \\ + \red{Two-point kernel} & & \violet{Four-point kernel} + \\ + $K(12) = v(12) + \pdv{V^{xc}(1)}{\rho(2)}$ & & $i \Xi(1234) = v(13) \delta(12) \delta(34) - \pdv{\Sigma^{xc}(12)}{G(34)}$ \\ + \hline + \end{tabular} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- + +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Relationship between CIS, TDHF, DFT and TDDFT} + \center + \begin{tikzpicture} + \usetikzlibrary{shapes.misc} + \begin{scope}[very thick, + node distance=3cm,on grid,>=stealth', + box/.style={rectangle,draw,fill=green!40}], + \node [box, align=center] (CIS) {\textbf{CIS}}; + \node [box, align=center] (HF) [left=of CIS, yshift=1cm] {\textbf{HF}}; + \node [box, align=center] (TDHF) [right=of CIS, yshift=1cm] {\textbf{TDHF}}; + \node [box, align=center] (DFT) [below=of HF] {\textbf{DFT}}; + \node [box, align=center] (TDDFT) [below=of TDHF] {\textbf{TDDFT}}; + \node [box, align=center] (TDA) [below=of CIS] {\textbf{TDA}}; + \path + (CIS) edge [<-] node[below,sloped]{CI} (HF) + (CIS) edge [<-] node[below,sloped]{$\bB{}{}=\bO$} (TDHF) + (HF) edge [->] node[above]{linear response} (TDHF) + (HF) edge [<->] node[left]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (DFT) + (TDHF) edge [<->] node[right]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (TDDFT) + (DFT) edge [->] node[above]{linear response} (TDDFT) + (DFT) edge [->] node[below,sloped]{CI} node[strike out,sloped]{\alert{$\cross$}} (TDA) + (TDDFT) edge [->] node[below,sloped]{$\bB{}{}=\bO{}{}$} (TDA) + ; + \end{scope} + \end{tikzpicture} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Useful papers} \begin{itemize} \item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149 \bigskip @@ -299,192 +931,18 @@ decoration={snake, \bigskip \item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102 \bigskip - \item \orange{Review:} Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344; Onida et al. Rev. Mod. Phys. 74 (2002) 601. + \item \orange{Review:} + \begin{itemize} + \item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344 + \item Onida et al. Rev. Mod. Phys. 74 (2002) 601 + \item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022 + \item Golze et al. Front. Chem. 7 (2019) 377 + \item Blase et al. JPCL 11 (2020) 7371 + \end{itemize} \bigskip \item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 \end{itemize} \end{frame} - %----------------------------------------------------- -\section{Implementation} -%----------------------------------------------------- -\subsection{\GOWO} -%----------------------------------------------------- -\begin{frame}{\GOWO} - \begin{block}{{\GOWO}~subroutine} - \begin{algorithmic} - \Procedure{Perturbative {\GW}}{} - \State Perform HF calculation to get $\beHF$ and $\bcHF$ - \For{$p=1,\ldots,N$} - \State Compute \red{$\SigC{pp}(\omega)$} and \green{$\Z{p}(\omega)$} - \State $\eGOWO{p} = \eHF{p} + \green{\Z{p}(\eHF{p})} \Re[\red{\SigC{pp}(\eHF{p})}]$ - \State \Comment{This is the linearized version of the} - \State \Comment{quasiparticle (QP) equation - $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$} - \EndFor - \If{BSE} - \State Compute BSE excitations energies - \EndIf - \EndProcedure - \end{algorithmic} - \end{block} -\end{frame} - -\begin{frame}{\GOWO} - \begin{block}{\red{Correlation part of the self-energy:}} - \begin{equation*} - \red{\SigC{pq}(\omega)} - = 2 \sum_{ix}\frac{\violet{[pi|x] [qi|x]}}{\omega - \eHF{i} + \orange{\Om{x}} - i \eta} - + 2 \sum_{ax}\frac{\violet{[pa|x] [qa|x]}}{\omega - \eHF{a} - \orange{\Om{x}} + i \eta} - \end{equation*} - \end{block} - \begin{block}{\green{Renormalization factor}} - \begin{equation*} - \green{\Z{p}(\omega)} = \qty[ 1 - \pdv{\Re[\red{\SigGW{pp}(\omega)}]}{\omega} ]^{-1} - \end{equation*} - \end{block} - \begin{block}{\violet{Screened two-electron MO integrals}} - \begin{equation*} - \violet{[pq|x]} = \sum_{ia} (pq|ia) \orange{(\bX+\bY)_{ia}^{x}} - \end{equation*} - \end{block} - \begin{block}{\orange{RPA excitation energies}} - \small - \begin{equation*} - \begin{pmatrix} - \bA & \bB \\ - \bB & \bA \\ - \end{pmatrix} - \orange{\begin{pmatrix} - \bX \\ - \bY \\ - \end{pmatrix}} - = - \orange{\bOm} - \begin{pmatrix} - \bm{1} & 0 \\ - 0 & \bm{-1} \\ - \end{pmatrix} - \orange{\begin{pmatrix} - \bX \\ - \bY \\ - \end{pmatrix}} - \end{equation*} - \begin{align*} - A^\text{RPA}_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb) - & - B^\text{RPA}_{ia,jb} & = 2 (ia|bj) - \end{align*} - \end{block} -\end{frame} - -%----------------------------------------------------- -\subsection{\evGW} -%----------------------------------------------------- -\begin{frame}{\evGW} - \begin{block}{{\evGW} subroutine} - \begin{algorithmic} - \Procedure{Partially self-consistent {\evGW}}{} - \State Perform HF calculation to get $\beHF$ and $\bcHF$ - \State Set $\beGnWn{-1} = \beHF$ and $n = 0$ - \While{$\max{\abs{\bDelta}} < \tau$} - \For{$p=1,\ldots,N$} - \State Compute \red{$\SigC{pp}(\omega)$} - \State Solve $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$ to obtain $\eGnWn{p}{n}$ - \EndFor - \State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$ - \State $n \leftarrow n + 1$ - \EndWhile - \If{BSE} - \State Compute BSE excitations energies - \EndIf - \EndProcedure - \end{algorithmic} - \end{block} -\end{frame} - -%----------------------------------------------------- -\subsection{\qsGW} -%----------------------------------------------------- -\begin{frame}{\qsGW} - \begin{block}{{\qsGW} subroutine} - \begin{algorithmic} - \Procedure{Partially self-consistent {\qsGW}}{} - \State Perform HF calculation to get $\beHF$ and $\bcHF$ - \State Set $\beGnWn{-1} = \beHF$, $\bcGnWn{-1} = \bcHF$ and $n = 0$ - \While{$\max{\abs{\bDelta}} < \tau$} - \State Form \red{$\bSigC(\omega)$} and symmetrize it: $\red{\bSigC(\omega)} \leftarrow (\red{\bSigC(\omega)}^\dag + \red{\bSigC(\omega)})/2$ - \State Form $\green{\bF(\omega)} = \bFHF + \red{\bSigC(\omega)}$ - \State Diagonalize $\green{\bF(\beGnWn{n-1})}$ to get $\beGnWn{n}$ and $\bcGnWn{n}$ - \State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$ - \State $n \leftarrow n + 1$ - \EndWhile - \If{BSE} - \State Compute BSE excitations energies - \EndIf - \EndProcedure - \end{algorithmic} - \end{block} -\end{frame} - -%----------------------------------------------------- -\subsection{BSE} -%----------------------------------------------------- -\begin{frame}{BSE} - \begin{block}{Bethe-Salpeter equation} - \begin{equation*} - \begin{pmatrix} - \bA & \bB \\ - \bB & \bA \\ - \end{pmatrix} - \purple{\begin{pmatrix} - \bX \\ - \bY \\ - \end{pmatrix}} - = - \purple{\bOm} - \begin{pmatrix} - \bm{1} & 0 \\ - 0 & \bm{-1} \\ - \end{pmatrix} - \purple{\begin{pmatrix} - \bX \\ - \bY \\ - \end{pmatrix}} - \end{equation*} - \begin{equation*} - (\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ, - \end{equation*} - \begin{equation*} - \bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ. - \end{equation*} - \begin{align*} - A^\text{BSE}_{ia,jb} & = A^\text{RPA}_{ia,jb} - (ij|ab) + 4 \sum_{x} \frac{[ij|x][ab|x]}{\Om{x}} - \\ - B^\text{BSE}_{ia,jb} & = B^\text{RPA}_{ia,jb} - (ib|aj) + 4 \sum_{x} \frac{[ib|x][aj|x]}{\Om{x}} - \end{align*} - \end{block} -\end{frame} - -%----------------------------------------------------- -\subsection{$\Ec$} -%----------------------------------------------------- -\begin{frame}{Correlation energy} - - \begin{block}{RPA correlation energy or Klein functional} - \begin{equation*} - \label{eq:Ec-RPA} - \EcRPA = -\sum_{p} \qty(\ARPA{pp} - \Om{p}) - \end{equation*} - \end{block} - - \begin{block}{Galitskii-Migdal functional} - \begin{equation*} - \label{eq:GM} - \EcGM = \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \SigC{pq}(\omega) \G{pq}(\omega) e^{i\omega\eta} - \end{equation*} - \end{block} - -\end{frame} \end{document} diff --git a/2021/Lecture_2/fig/BSE-GW.tex b/2021/Lecture_2/fig/BSE-GW.tex index f5cda29..e196337 100644 --- a/2021/Lecture_2/fig/BSE-GW.tex +++ b/2021/Lecture_2/fig/BSE-GW.tex @@ -36,18 +36,18 @@ {\textbf{\LARGE Bethe-Salpeter equation} $$ \begin{pmatrix} - R & C \\ - -C^* & -R^{*} + \bm{A} & \bm{B} \\ + -\bm{B}^* & -\bm{A}^{*} \end{pmatrix} \begin{pmatrix} - X_m \\ - Y_m + \bm{X}_m \\ + \bm{Y}_m \end{pmatrix} = \Omega_{m} \begin{pmatrix} - X_m \\ - Y_m + \bm{X}_m \\ + \bm{Y}_m \end{pmatrix} $$ };